We generalise results of Ford and Roman which place lower bounds -known as quantum inequalities -on the renormalised energy density of a quantum field averaged against a choice of sampling function. Ford and Roman derived their results for a specific non-compactly supported sampling function; here we use a different argument to obtain quantum inequalities for a class of smooth, even and non-negative sampling functions which are either compactly supported or decay rapidly at infinity. Our results hold in d-dimensional Minkowski space (d ≥ 2) for the free real scalar field of mass m ≥ 0. We discuss various features of our bounds in 2 and 4 dimensions. In particular, for massless field theory in 2-dimensional Minkowski space, we show that our quantum inequality is weaker than Flanagan's optimal bound by a factor of 3 2 .

Abstract. We study a phenomenon occuring in various areas of quantum physics, in which an observable density (such as an energy density) which is classically pointwise nonnegative may assume arbitrarily negative expectation values after quantisation, even though the spatially integrated density remains nonnegative. Two prominent examples which have previously been studied are the energy density (in quantum field theory) and the probability flux of rightwardsmoving particles (in quantum mechanics). However, in the quantum field context, it has been shown that the magnitude and space-time extension of negative energy densities are not arbitrary, but restricted by relations which have come to be known as 'quantum inequalities'. In the present work, we explore the extent to which such quantum inequalities hold for typical quantum mechanical systems. We derive quantum inequalities of two types. The first are 'kinematical' quantum inequalities where spatially averaged densities are shown to be bounded below. Specifically, we obtain such kinematical quantum inequalities for the current density in one spatial dimension (imposing constraints on the backflow phenomenon) and for the densities arising in Weyl-Wigner quantization. The latter quantum inequalities are direct consequences of sharp Gårding inequalities. The second type are 'dynamical' quantum inequalities where one obtains bounds from below on temporally averaged densities. We derive such quantum inequalities in the case of the energy density in general quantum mechanical systems having suitable decay properties on the negative spectral axis of the total energy.Furthermore, we obtain explicit numerical values for the quantum inequalities on the onedimensional current density, using various spatial averaging weight functions. We also improve the numerical value of the related 'backflow constant' previously investigated by Bracken and Melloy. In many cases our numerical results are controlled by rigorous error estimates.

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This paper may be regarded as a sequel to our earlier paper [19], where we give an elementary and self-contained proof of a very general form of the Hopf theorem on order-preserving linear operators in partially ordered vector spaces (reproduced here as Theorem 1·1).Versions of this theorem and related ideas have been used by various authors to study both linear and nonlinear integral equations (Thompson [41], Bushell [9, 11], Potter [38, 39], Eveson [16, 17], Bushell and Okrasiriski [12, 13]); the convergence properties of nonlinear maps (Nussbaum [32, 33]); so-called DAD theorems (Borwein, Lewis and Nussbaum [8]) and in the proof of weak ergodic theorems (Fujimoto and Krause [20], Nussbaum [34]).

\\L n x\\ * Partially supported by NSFDMS 91-05930. 32 S I M O N P . E V E S O N AND R O G E R D. NUSSBATJMfor all positive vectors x. In addition, one may obtain explicitly computable formulae for the so-called spectral clearance q(L) given by where W with L(C) <~D. Our basic observation is that it suffices to prove the theorem in the case when V and W are two-dimensional. Next we analyse two-dimensional cones and show that it suffices to prove the theorem when V = W= U 2 , C =D = {xsU 2 :x 1 ,x 2^0 }&ndL = ft ^], with a > 1. Proving the theorem in this case is a simple calculus exercise which is carried out in Section 5. An amusing benefit of our proof is that, in contrast to all previous work, we need no assumption that our cones are Archimedean or almost Archimedean.The approach here basically follows unpublished notes of R. D. Nussbaum which were written in 1986-87 and were one topic in a series of lectures at Emory University in the Spring of 1988. Independently, S. P. Eveson [12,11] found a closely related proof of the theorem for the almost Archimedean case. The present paper unites and refines these two approaches. Preliminary definitions and resultsDefinition 2-1. If V is a real vector space and C is a subset of V, we shall call C a cone (with vertex at 0) if it satisfies the following three properties:(1) C is convex;(2) tC £ C whenever t ^ 0, where tO = {tx: xeC};(3) CO (-C) = {0}. If C satisfies properties (1) and (2), but not necessarily (3), we shall call C a wedge.Remark 2-2. Note that we do not, in contrast to some of the literature, assume that V is a topological v...